Fractal and multifractal descriptors restore ergodicity broken by non-Gaussianity in time series

TL;DR

This paper demonstrates that fractal and multifractal descriptors can restore ergodicity in long-range correlated non-Gaussian time series, improving causal modeling in biological and psychological sciences.

Contribution

It introduces fractal and multifractal descriptors as tools to restore ergodicity in non-Gaussian, long-range correlated time series, addressing a key challenge in causal modeling.

Findings

Fractal and multifractal descriptors do not exhibit ergodicity breaking.

Non-Gaussianity and long-range correlations jointly break ergodicity in standard measures.

Fractal/multifractal descriptors restore ergodicity in these processes.

Abstract

Ergodicity breaking is a challenge for biological and psychological sciences. Ergodicity is a necessary condition for linear causal modeling. Long-range correlations and non-Gaussianity characterizing various biological and psychological measurements break ergodicity routinely, threatening our capacity for causal modeling. Long-range correlations (e.g., in fractional Gaussian noise, a.k.a. "pink noise") break ergodicity--in raw Gaussian series, as well as in some but not all standard descriptors of variability, i.e., in coefficient of variation (CV) and root mean square (RMS) but not standard deviation (SD) for longer series. The present work demonstrates that progressive increases in non-Gaussianity conspire with long-range correlations to break ergodicity in SD for all series lengths. Meanwhile, explicitly encoding the cascade dynamics that can generate temporally correlated non-Gaussian noise offers a way to restore ergodicity to our causal models. Specifically, fractal and multifractal properties encode both scale-invariant power-law correlations and their variety, respectively, both of which features index the underlying cascade parameters. Fractal and multifractal descriptors of long-range correlated non-Gaussian processes show no ergodicity breaking and hence, provide a more stable explanation for the long-range correlated non-Gaussian form of biological and psychological processes. Fractal and multifractal descriptors offer a path to restoring ergodicity to causal modeling in these fields.